Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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September 8, 2023 at 9:26am

#1055360

It's All One, Dude

I've written about this sort of thing before, but it's been a while, and this is a different article. For reference, my last bit on the topic of the superficial similarity of galactic and neural networks was here: "Networking"

As for today's article, which I have different things to say about:

The case for why our Universe may be a giant neural network
Neuroscientist and author Bobby Azarian explores the idea that the Universe is a self-organizing system that evolves and learns.

It is natural, I think, for humans to notice similarities among disparate things, and, once noting similarity of form, speculate about a corresponding similarity of function. It's part of the basis for some spiritual systems, as well as poetry and metaphor. The trap we don't want to fall into, though, would be (in this case) assuming that such speculation reflects reality, without much evidence beyond, "Well, they look the same."

A new scientific paradigm is emerging that presents us with a radically different cosmic narrative. The big idea is that the Universe is not just an arbitrary physical system, but something more like an evolving computational or biological system — with properties strikingly similar to a complex adaptive system, like an organism or a brain.

Calling it a "paradigm" is overstating the case, I think. I should note, as the article does, that the idea that the cosmos is a kind of entity in itself is nothing new; that, too, is part of ancient spiritual systems. What's different is that, more recently, within the last 100 years or so, our knowledge of the size and scale of the universe has expanded (pun intended).

If true, it raises new existential questions that will force us to completely rethink the nature of reality and ideas about whether the Universe has a function or “purpose.”

"If" does a whole lot of lifting, there.

In recent years, a number of highly respected theoretical physicists and scientists from various fields have published papers, articles, and books that have provided compelling technical and mathematical arguments that suggest the Universe is not just a computational or information-processing system, but a self-organizing system that evolves and learns in ways that are strikingly similar to biological systems.

I'd just like to point out two things here. First, this borders on "argument from authority;" even "highly respected" scientists can be, and often are, wrong. And second, there doesn't have to be anything mystical or supernatural about this. Lots of things, as I noted above, are similar to each other, as they follow the same universal laws.

For example, scientists have recently emphasized that the physical organization of the Universe mirrors the structure of a brain.

As the universe is several orders of magnitude older than a human brain (or any sort of brain that we know of), I'd say that should be the other way around.

Theoretical physicist Sabine Hossenfelder — renowned for her skepticism — wrote a bold article for Time Magazine in August of 2022 titled “Maybe the Universe Thinks. Hear Me Out,” which describes the similarities.

Okay, I admit I have a thing for Dr. Hossenfelder. No, not like that; I just think she's remarkably good at science communication. I've seen a bunch of her videos and I've read a couple of her books. As with anyone else, though, she can still be wrong. And her "skepticism" takes a different form than most public scientists'—she likes to make distinctions between "consistent with science" and "borne out by evidence," which I appreciate.

For instance, and I'm not saying she tackled this subject, Bigfoot is consistent with science: a living entity without supernatural powers who, if real (it isn't), is just remarkably good at hiding in the woods. There is still no evidence for Bigfoot's existence, and some evidence that it's completely fabricated.The Jersey Devil, however, is not consistent with science.

That said, I haven't read the article mentioned here.

When one zooms out to envision the cosmos as a whole, the “cosmic web” formed by these clusters and filaments looks strikingly similar to the “connectome,” a term that refers to the complete wiring diagram of the brain, which is formed by neurons and their synaptic connections. Neurons in the brain also form clusters, which are grouped into larger clusters, and are connected by filaments called axons, which transmit electrical signals across the cognitive system.

I noticed this similarity of appearance many years ago (and wrote about it in here before), but again: that doesn't imply similarity of function.

Hossenfelder explains that this resemblance between the cosmic web and the connectome is not superficial, citing a rigorous study by a physicist and a neuroscientist that analyzed the features common to both, and based on the shared mathematical properties, concluded that the two structures are “remarkably similar.”

Fortunately, this article does go into some of the arguments against the idea.

Of course, it takes more than a certain type of structure to think thoughts. A dead brain is just as thoughtless as a rock.

and

Yet the Universe’s vastness imposes limitations. Hossenfelder explains that sending signals across the cosmos, even at light speed, would take 80 billion years, and 11 million years just for a signal to travel to our nearest galaxy. Combine the vast size of the Universe with the fact that it is expanding, and it would seem like some kind of cosmic-scale information processing similar to the global processing going on inside brains is out of the question.

However,

But Hossenfelder speculates over whether “hidden connections” could allow for faster signaling. In a section called “Everything is Connected,” she explains how mechanisms like quantum entanglement or other forms of “non-local connections” could enable longer-range computations.

This demonstrates either a remarkable lapse in communication skills (for her or for the author of the article here), or a misunderstanding of how quantum entanglement really works. I tend to believe the former, as I've seen her expand upon the latter.

And let's just emphasize this throwaway line:

Although highly speculative...

So the rest of the article is worth reading, but I won't bore you with more of my specific thoughts about it; they're all along the same lines. I'll just add one more thing for today:

The thing is, I would like to believe that everything is interconnected. It suits my philosophy of life. On a purely local level, meaning at about the scale of the solar system or below, this is so obvious as to be almost trivial; it's only the philosophical implications that reach into the depths of the profound. We have a pretty good understanding of the forces that make it so: gravity, electromagnetism, etc. Pull one thread, and others shiver. No man is an island, etc. (And even if one was, the whole concept of "island" is defined by the surrounding water.)

The only remarkable thing is that I didn't use drugs to come to this conclusion.

It's just that, when I encounter speculation that I would like to believe, my personal threshold for evidence gets higher, not lower.

The article says some people think this is testable.

So test it.

I'd love to be right. But I'm not counting on it.

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